# fixed point

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Related to fixed point: Fixed point number, Fixed point theorem

## fixed point

[¦fikst ′pȯint]
(engineering)
A reproducible value, as for temperature, used to standardize measurements; derived from intrinsic properties of pure substances.
(mathematics)
For a function ƒ mapping a set S to itself, any element of S which ƒ sends to itself.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Fixed Point

a form of representation of numbers in a digital computer with constant position of the point that sepa-rates the whole part of the number from the fraction. The fixed point corresponds to the natural form of representation of numbers. The point may be fixed at any position of the number—for example, in a digital computer five-place numbers with a fixed point after the second place are represented as +74.531, +07.453, +00.745, and so on. To prevent the numbers formed in the process of calculations from going beyond the range of representable numbers, scale factors are incorporated into the input data and intermediate and final results when drawing up programs for computers with a fixed point. However, the fixing of the point before the high-order digit of the modulus of the number (a number less than 1) is more expedient; in such a case the word format of the digital computer is not overloaded during multiplication of numbers. The range of representable numbers is narrower in a digital computer with a fixed point than in a digital computer with a floating point. The complication of programming when a fixed point is used is compensated in some cases by the simplicity of the devices of the digital computer and the ease in carrying out arithmetic operations, and also by the possibility of achieving greater speed in addition and subtraction. A fixed point was used in the Soviet Minsk-1, Setun’, and Ural-1 digital computers and in most digital control computers.

A. V. GUSEV

## fixed point

(mathematics)
The fixed point of a function, f is any value, x for which f x = x. A function may have any number of fixed points from none (e.g. f x = x+1) to infinitely many (e.g. f x = x). The fixed point combinator, written as either "fix" or "Y" will return the fixed point of a function.

## fixed point

A method for storing and calculating numbers in which the decimal point is always in the same location. Contrast with floating point.
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References in periodicals archive ?
It follows that the fixed point of [psi] is unique.
For [epsilon] > 0, there exists [delta](= [epsilon]) such that all the hypotheses of Theorem 9 are valid to obtain common fixed point of [psi] and [PHI].
The mapping T satisfies all others conditions of Theorem 10 but has no fixed point in X.
Then T has a unique fixed point, and for every initial point [x.sub.0] [member of] X, the Picard sequence {[T.sup.n][x.sub.0]} converges to the fixed point.
Then [g.sup.*] is a fixed point of B in [mathematical expression not reproducible].
Similarly as in Theorem 20 by using condition (26) and property [mathematical expression not reproducible] and weak compatibility it can be shown that z is a unique common fixed point.
Since all eigenvalues of M are complex numbers with modulus less than 1, then (p, p) is a stable fixed point; see [13, 14].
Kolahi, Nielsen coincidence, fixed point and root theories of n-valued maps.
Section 3 deals with convex-valued multimaps in the KKM theory and analytical fixed point theory, that is, one of the most important applications of the KKM theory.
Near hyperbolic fixed point, a nonlinear dynamical system could be linearized and stability of the fixed point is found by Hartman-Grobman theorem.
Computational results are summarized in Table 1, where [x.sup.(0)] denotes the initial guess, IT is the number of iterations, H is the value of [mathematical expression not reproducible] when the algorithm stops, and [x.sup.*] is the fixed point.
The basin of attraction of an attracting fixed point x*, A(x*), is defined as the set of preimages of any order such that

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